Modeling and closed-loop displacement control method for shape memory alloy driver

Abstract

The invention discloses a modeling and closed-loop displacement control method for a shape memory alloy driver, and the method comprises the steps: taking an inverse hysteresis feed-forward compensation and SMC-PID feedback displacement control method as a core, building a positive model of the shape memory alloy driver, carrying out the inversion to obtain an inverse model, designing a self-adaptive SMC-PID feedback controller, and carrying out the closed-loop displacement control of the shape memory alloy driver. Combining the positive model, the inverse model and an SMC-PID feedback controller to construct a control system for closed-loop displacement control of the shape memory alloy driver; matlab / Simulink software is used for carrying out modeling simulation testing on the designed control system, and the closed-loop displacement control effect of the shape memory alloy driver is observed by selecting different tracking signals. The modeling and closed-loop displacement control method for the shape memory alloy driver has an excellent control effect under various motion working conditions such as back-and-forth reciprocation, continuous heating deformation and continuous cooling deformation of the shape memory alloy driver, is high in tracking precision and small in delay, and realizes high robustness for system delay and parameter uncertainty.

Description

Technical Field

[0001] The present invention relates to the field of intelligent material and mechanism modeling and control, and in particular to a modeling and closed-loop displacement control method of a shape memory alloy driver. Background Art

[0002] Shape memory alloys (SMA) have unique shape memory effects, superelasticity, high damping, and self-sensing properties. Shape memory alloy actuators designed based on these properties have high power-to-weight ratio, high strain stress, high drive frequency, and other characteristics. They have been widely studied and applied in aerospace, automation, and robotics. For example, in the aerospace field, shape memory alloy actuators are widely used in various drive mechanisms, such as satellite sailboard deployment mechanisms, variable wing control devices, fastening and unlocking mechanisms, engine intake and exhaust adjustment mechanisms, etc. In recent years, with the increase in application scenarios, the demand for precise position control of shape memory alloy actuators has become increasingly prominent, and precise position control has become a research hotspot at home and abroad.

[0003] The action of the shape memory alloy actuator originates from the phase change process of SMA, which is reversible and has the characteristics of high nonlinearity, large hysteresis and slow cooling. The traditional "open-loop operation" control method has limited the further development of the shape memory alloy actuator. Closed-loop control is the main control form of displacement control of shape memory alloy actuators at this stage. Therefore, in order to solve the problem of complex hysteresis nonlinearity of the displacement of shape memory alloy actuators and realize higher precision displacement control of shape memory alloy actuators, it is necessary to propose more effective control strategies and design controllers with better performance. Summary of the invention

[0004] In view of the large hysteresis nonlinearity of shape memory alloys, in order to improve the displacement control accuracy and robustness of shape memory alloy actuators, the present invention proposes a modeling and closed-loop displacement control method for shape memory alloy actuators.

[0005] In order to achieve the above object, the present invention adopts the following technical solution:

[0006] A modeling and closed-loop displacement control method for a shape memory alloy actuator comprises the following steps:

[0007] S1: A mechanism model of the shape memory alloy actuator is established based on the working principle of the shape memory alloy actuator, and a mathematical model of the shape memory alloy actuator is established in a modular manner according to the mechanism model, namely, a positive model. The positive model takes current as input and shape memory alloy wire strain as output, and includes a heat transfer and temperature module, a phase change hysteresis module, a strain module, and a resistance module;

[0008] S2: using the forward model to invert and establish an inverse model of the shape memory alloy actuator, the inverse model is derived in a modular manner, with the desired strain as input and the required current calculated by the inverse model as output. Through the inverse model of the shape memory alloy actuator, the hysteresis effects of the highly nonlinear, large hysteresis and slow cooling caused by the phase change process inside the shape memory alloy will be compensated before the shape memory alloy actuator encounters them, and the inverse model is thus used for inverse hysteresis feedforward compensation control;

[0009] S3: Design an adaptive sliding mode controller based on proportional-integral-derivative tuning as the feedback control part for realizing precise displacement control of the shape memory alloy actuator, where the control signal consists of a sliding control signal and a sliding mode equivalent control signal. A proportional-integral-derivative tuning controller is used to approximate the sliding mode equivalent control signal, and Lyapunov's law is used to prove the stability of the closed-loop feedback control system.

[0010] S4: combining the forward model described in step S1, the inverse model described in step S2, and the adaptive sliding mode controller based on proportional-integral-derivative tuning described in step S3 to construct a control system for closed-loop displacement control of the shape memory alloy actuator;

[0011] S5: Use Matlab / Simulink software to simulate and test the control system described in step S4, and observe the closed-loop displacement control effect of the shape memory alloy actuator by selecting different tracking signals.

[0012] Further, step S1 includes:

[0013] S1.1: The shape memory alloy actuator is driven by a spring bias. The shape memory alloy actuator is composed of a bias spring and a shape memory alloy wire. One end of the spring and the shape memory alloy wire are fixed at the fixed end respectively. When the shape memory alloy wire is powered on and heated, its material contracts due to the shape memory effect and generates tension. When the tension exceeds the elastic force of the spring, the displacement of the end of the shape memory alloy wire will change accordingly. When cooled, the shape memory alloy wire expands with the help of the bias spring. The purpose of accurately controlling the displacement of the shape memory alloy actuator is achieved by accurately controlling the current input to the shape memory alloy wire.

[0014] S1.2: The heat transfer and temperature module is used to describe the heat transfer problem of the shape memory alloy wire in the shape memory alloy actuator. The heat transfer and temperature module model is given by the following expression (1), with current as input and the temperature of the shape memory alloy wire as output:

[0015]

[0016] Where R is the internal resistance of the alloy wire, d0 and L0 are the cross-sectional diameter and length of the undeformed alloy wire, respectively, and T amb is the ambient temperature, ρ is the mass density of the alloy wire, c is the specific heat of the alloy wire, h is the convection heat transfer coefficient of the alloy wire, c and h are functions of the temperature T;

[0017] S1.3: The phase change hysteresis module is used to describe the martensite volume fraction R of the shape memory alloy wire in the shape memory alloy actuator. m The relationship with temperature T is given by the Duhem differential hysteresis model. The phase change hysteresis model is given by the following expression (2):

[0018]

[0019] Among them, g + / - is the Gaussian probability density function, describing the slope of the hysteresis loop, h + / - It is for g + / - The integral of describes the shape of the hysteresis loop. The subscripts + and - represent increasing and decreasing curves, respectively. μ is the mean value, and σ 2 is the variance, which is used to control the shape of the hysteresis loop. is the error function when integrating the normal distribution;

[0020] S1.4: The strain module is used to describe the strain ε and martensite volume fraction R of the shape memory alloy wire in the shape memory alloy actuator. m The relationship between R and ε is given by the following formula (3). The total strain ε can be approximated as m Polynomial of:

[0021]

[0022] Among them, ε0 is the strain caused by the preload of the alloy wire in the fully austenitic state, and the values ​​of k1, k2, and k3 are related to the specific shape memory alloy wire;

[0023] S1.5: The resistance module is used to describe the resistance of the shape memory alloy wire in the shape memory alloy actuator, and is provided to step S1.2 for calculation, which is given by the following formula (4): the temperature T and the martensite volume fraction R calculated by steps S1.2, S1.3 and S1.4 are: m The resistance of the shape memory alloy wire can be calculated by the strain ε:

[0024]

[0025] Among them, ρ a (T) is the resistivity of austenite, ρ m (T) is the resistivity of martensite, p1, p2, p3, q1, q2 and α i(i=1-9) is the correlation constant.

[0026] Further, step S2 includes:

[0027] Assume the desired strain trajectory is ε rd (t), according to the forward model of the shape memory alloy actuator, the inverse model of the strain module, the phase change hysteresis module, the heat transfer and temperature module and the resistance module are obtained by inverting in sequence, as shown in the following formula (5), thereby obtaining the inverse model of the shape memory alloy actuator whose input is the desired strain and whose output is the required current calculated by the inverse model. The inverse model is derived from the highly nonlinear, large hysteresis and slow cooling hysteresis effects of the phase change process inside the shape memory alloy, and the inverse model is thus used for inverse hysteresis feedforward compensation control;

[0028]

[0029] Among them, δ>0 is an arbitrarily small constant to avoid calculation overflow caused by the denominator being zero.

[0030] Further, step S3 includes:

[0031] Design an adaptive sliding mode controller based on proportional-integral-derivative tuning. The control variable u is composed of the sliding control signal and the sliding mode equivalent control signal: u = u s +u eq , where u s is the sliding control signal, which switches the system to the sliding surface; u eq It is a sliding mode equivalent control; a proportional-integral-differential tuning controller is used to approximate the sliding mode equivalent control signal, and Lyapunov's law is used to prove the stability of the closed-loop feedback control system. The specific design process is:

[0032] S3.1: Mathematical description of shape memory alloy actuators:

[0033]

[0034] Where x = [x1(t), x2(t)] T represents the displacement and velocity of the end of the shape memory alloy wire in the shape memory alloy actuator; b(x1, x2) is the input gain, which represents the internal resistance of the shape memory alloy; f(·) is a nominal parameter of the model, Δf(·) represents the phase change of the shape memory alloy, describing the hysteresis phenomenon of the shape memory alloy; d(t) represents the external disturbance; u(t) = I is the input. Δf(·), d(t) and b(·) are all bounded variables, |Δf(·)|≤g, d(t)|≤α, b min ≤b(·)≤b max ; g is the maximum change of the hysteresis curve; α is the maximum value of the disturbance;

[0035] S3.2: Design of integral sliding surface σ

[0036] In order to improve the convergence performance of the sliding surface, the reference signal x is defined r , so that:

[0037]

[0038] Where e represents the expected displacement y d The error with the output y, e = y d -y; K1 and K0 make s 2 The root of +K1s+K0=0 is in the left half plane of the s plane;

[0039] Design integral sliding surface σ=x r -x2, substituting into equation (7), σ can be rewritten as When sliding mode occurs, σ tends to zero, and the introduction of integral links and high-order forms in the sliding mode function can effectively improve the control performance of the system;

[0040] S3.3: Stability analysis of closed-loop feedback control systems:

[0041] Define the Lyapunov function as V = 1 / 2σ 2 ,but To achieve system stability Design control u to satisfy K>0; combined with formula (6), the sliding surface σ=x r The derivative of -x2 can be expressed as:

[0042]

[0043] Therefore, the control quantity is obtained as:

[0044]

[0045] The control quantity u can be divided into two parts: u = u s +u eq .u s is the sliding control signal, which switches the system to the sliding surface. u eq is the sliding mode equivalent control,

[0046] S3.4: Sliding mode equivalent control signal u eq Design:

[0047] Use PID tuning controller to approximate equivalent control:

[0048]

[0049] For the PID parameter setting, the following objective function is selected: And use the gradient descent method to obtain the update rule of PID parameters:

[0050]

[0051] Since when ε u →0, there exists σ→0, so the above PID parameter update rule can be optimized as:

[0052]

[0053] S3.5: Sliding control signal u s Design:

[0054] From equations (8), (9) and (10), we can deduce:

[0055]

[0056] Among them, u PID is a positive term, so if K is selected to satisfy the following conditions, then the system must be asymptotically stable If and only if σ=0, the equality holds:

[0057]

[0058] In order to weaken the high-frequency chattering inherent in the control input, the saturation function sat(σ / δ) is selected to replace sgn(σ), where δ is the width of the boundary layer. The sliding surface function σ with any initial value will reach and remain within the boundary layer σ|<δ, and the sliding control signal u s becomes:

[0059]

[0060] Further, step S4 includes:

[0061] S4.1: The output of the inverse model of the shape memory alloy actuator established in step S2 is: the current required by the shape memory alloy actuator to achieve the desired strain, which is input into the forward model as a feedforward signal in the control system to compensate for the effects of the highly nonlinear, large hysteresis and slow cooling hysteresis of the phase change process inside the shape memory alloy on the shape memory alloy actuator;

[0062] S4.2: In step S3, an adaptive sliding mode controller based on proportional-integral-derivative tuning is designed. The control variable u is composed of the sliding control signal and the sliding mode equivalent control signal: u = u s +u eq ; This part is input into the mathematical model of the shape memory alloy actuator as a feedback control signal;

[0063] S4.3: Combine the forward model described in step S1, the inverse model described in step S2, and the adaptive sliding mode controller based on proportional-integral-derivative setting described in step S3 to construct a control system for closed-loop displacement control of the shape memory alloy actuator, which is used to accurately control the displacement of the shape memory alloy actuator.

[0064] Further, step S5 includes:

[0065] S5.1: According to the derivation of steps S1, S2, S3, and S4, a Simulink simulation platform is built;

[0066] S5.2: The input is a sinusoidal signal, simulating the sinusoidal tracking condition of the shape memory alloy actuator, and observing the tracking of the output displacement of the shape memory alloy actuator;

[0067] S5.3: The input is a continuous descending step signal, simulating the tracking condition of the shape memory alloy to the step command signal during the heating and contraction process, and observing the tracking condition of the output displacement of the shape memory alloy driver;

[0068] S5.4: The input is a continuous rising step signal, which simulates the tracking condition of the shape memory alloy to the step command signal during the recovery process, and observes the tracking condition of the output displacement of the shape memory alloy driver.

[0069] Compared with the prior art, the present invention has the following beneficial effects:

[0070] The control scheme proposed by the present invention has a simple structure and includes two parts: feedforward and feedback. The feedforward part is implemented based on the inverse model. The required inverse model is obtained by inverting each module of the mathematical model of the shape memory alloy driver, which is easy to obtain; the feedback part adopts a proportional-integral-differential (PID) parameter adaptive sliding mode controller (SMC), which can update the PID gain parameter online through the adaptive law, and use the PID controller to approximate the sliding mode equivalent control signal, weaken the high-frequency chattering of the sliding mode, and the feedback part control rate is simple in form.

[0071] The designed control method has better control effect than the traditional control method under various motion conditions such as reciprocating movement of shape memory alloy actuators, continuous heating deformation, and continuous cooling deformation. It has high tracking accuracy and small delay, and achieves strong robustness against system hysteresis and parameter uncertainty. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] The above and / or additional aspects and advantages of the present invention will become apparent and easily understood from the description of the embodiments in conjunction with the following drawings, in which:

[0073] Figure 1The present invention is a flow chart of a displacement control method based on inverse hysteresis feedforward compensation and SMC-PID feedback in an embodiment of the present invention, which realizes a displacement control process in which a driver is in the form of a shape memory alloy wire-spring;

[0074] Figure 2 This is a schematic diagram of the shape memory alloy actuator mechanism model;

[0075] Figure 3 A block diagram of a mathematical model (positive model) of a shape memory alloy actuator provided by an embodiment of the present invention;

[0076] Figure 4 A block diagram of an inverse hysteresis model of a shape memory alloy actuator provided by an embodiment of the present invention;

[0077] Figure 5 A control block diagram of a shape memory alloy driver combining feedforward, inverse hysteresis and SMC-PID feedback provided by an embodiment of the present invention;

[0078] Figure 6 A displacement tracking curve of a shape memory alloy driver provided by an embodiment of the present invention to a continuous sinusoidal command signal;

[0079] Figure 7 A displacement tracking curve of a shape memory alloy actuator provided by an embodiment of the present invention in response to a step command signal during a heating and contraction process;

[0080] Figure 8 The present invention provides a shape memory alloy actuator provided by an embodiment of the present invention, which tracks the displacement of a step command signal during the recovery (elongation) process. DETAILED DESCRIPTION

[0081] In order to make the purpose, technical scheme and advantages of the present invention clearer, the present invention will be described in more detail below in combination with theoretical formulas, drawings and specific embodiments. It should be noted that the described embodiments are only part of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0082] The present invention provides a modeling and closed-loop displacement control method for a shape memory alloy actuator. Figure 1 As shown, in the embodiment of the present invention, the displacement control method with inverse hysteresis feedforward compensation and adaptive sliding mode controller (SMC-PID) feedback based on proportional-integral-differential setting is taken as the core to realize a displacement control in which the driver is in the form of shape memory alloy wire-spring, and the method includes:

[0083] S1: A mechanism model of the shape memory alloy actuator is established based on the working principle of the shape memory alloy actuator, and a mathematical model of the shape memory alloy actuator is established in a modular manner according to the mechanism model, namely, a positive model. The positive model takes current as input and shape memory alloy wire strain as output, and includes a heat transfer and temperature module, a phase change hysteresis module, a strain module, and a resistance module;

[0084] S2: The inverse model of the shape memory alloy actuator is established by inverting the forward model. The inverse model is derived in a modular way, with the desired strain as input and the required current calculated by the inverse model as output. Through the inverse model of the shape memory alloy actuator, the hysteresis effects of the highly nonlinear, large hysteresis and slow cooling caused by the phase change process inside the shape memory alloy will be compensated before the shape memory alloy actuator encounters them. The inverse model is thus used for inverse hysteresis feedforward compensation control;

[0085] S3: Design an adaptive sliding mode controller based on proportional-integral-derivative tuning as the feedback control part for realizing precise displacement control of the shape memory alloy actuator, where the control signal consists of a sliding control signal and a sliding mode equivalent control signal. A proportional-integral-derivative tuning controller is used to approximate the sliding mode equivalent control signal, and Lyapunov's law is used to prove the stability of the closed-loop feedback control system.

[0086] S4: combining the forward model in step S1, the inverse model in step S2, and the adaptive sliding mode controller based on proportional-integral-derivative tuning in step S3 to construct a control system for closed-loop displacement control of the shape memory alloy actuator;

[0087] S5: Use Matlab / Simulink software to simulate and test the control system in step S4, and observe the closed-loop displacement control effect of the shape memory alloy actuator by selecting different tracking signals.

[0088] Wherein, step S1 specifically includes the following steps:

[0089] S1.1: The shape memory alloy actuator is driven by a spring bias. The working principle is simplified as follows: Figure 2 As shown. The shape memory alloy actuator is composed of a bias spring and a shape memory alloy wire. One end of the spring and the shape memory alloy wire are fixed at the fixed end respectively. When the shape memory alloy wire is powered on and heated, its material contracts due to the shape memory effect and generates tension. When the tension exceeds the elastic force of the spring, the displacement of the end of the shape memory alloy wire will change accordingly. When cooling, the shape memory alloy wire expands with the help of the bias spring. The purpose of accurately controlling the displacement of the shape memory alloy actuator is achieved by accurately controlling the current input to the shape memory alloy wire. Figure 2The alloy wire is heated to a completely austenitic state. L0 is the length of the undeformed alloy wire, ε0 is the strain caused by the preload of the alloy wire in the completely austenitic state, k is the elastic constant of the bias spring, Δ is the spring deformation in the completely austenitic state, ε r It is the strain caused by the alloy wire during the transformation from austenite to martensite;

[0090] S1.2: The heat transfer and temperature module describes the heat transfer problem of the shape memory alloy wire in the shape memory alloy actuator. The heat transfer and temperature module model is given by the following expression, with current as input and the temperature of the shape memory alloy wire as output:

[0091]

[0092] Where R is the internal resistance of the alloy wire, which is calculated by the resistance module in step S1.5, d0 and L0 are the cross-sectional diameter and length of the undeformed alloy wire, respectively, and T amb is the ambient temperature, ρ is the mass density of the alloy wire, c is the specific heat of the alloy wire, and h is the convection heat transfer coefficient of the alloy wire. c and h are functions of temperature T, and each parameter is a constant obtained by experimental calibration and fitting.

[0093] S1.3: Phase transformation hysteresis module describes the martensite volume fraction R of the shape memory alloy wire in the shape memory alloy actuator. m The relationship with temperature T is given by the Duhem differential hysteresis model, and the phase change hysteresis model is given by the following expression:

[0094]

[0095] Among them, g + / - is a Gaussian probability density function describing the slope of the hysteresis loop. + / - It is for g + / - The integral of describes the shape of the hysteresis loop. The subscripts + and - represent increasing and decreasing curves, respectively, μ is the mean, and σ 2 is the variance, which is used to control the shape of the hysteresis loop. is the error function when integrating the normal distribution.

[0096] S1.4: The strain module describes the relationship between the strain ε and the martensite volume fraction R of the shape memory alloy wire in the shape memory alloy actuator. m The relationship between R and ε is given by the following formula: The total strain ε can be approximated as m Polynomial of:

[0097]

[0098] Among them, ε0 is the strain caused by the preload of the alloy wire in the fully austenitic state, and the values ​​of k1, k2, and k3 are related to the specific shape memory alloy wire and can be obtained through parameter identification.

[0099] S1.5: The resistance module describes the resistance of the shape memory alloy wire in the shape memory alloy actuator, which is used to provide step S1.2 for calculation. The specific formula is given by the following formula: the temperature T calculated by steps S1.2, S1.3 and S1.4, the martensite volume fraction R m The resistance of the shape memory alloy wire can be calculated by the strain ε:

[0100]

[0101] Among them, ρ a (T) is the resistivity of austenite, ρ m (T) is the resistivity of martensite. p1, p2, p3, q1, q2 and α i (i=1-9) is the correlation constant.

[0102] In step S2, the inverse model of each module is obtained by inverting the forward model in S1, which is used as a feedforward to compensate for the nonlinearity and hysteresis of the shape memory alloy actuator. The specific process is:

[0103] Similar to the positive model structure in S1, the inverse model of the shape memory alloy actuator is derived in a modular way, as shown in the block diagram. Figure 4 Assume that the desired strain trajectory is ε rd (t), according to the modular mathematical model (forward model) of the shape memory alloy actuator, the inverse model of each module of the shape memory alloy actuator can be obtained in sequence, as shown in the following formula, which describes the inverse models of the strain module, phase change hysteresis module, heat transfer and temperature module and resistance module in sequence. Thus, the inverse model of the shape memory alloy actuator with the desired strain as input and the required current calculated by the inverse model as output is obtained, which is used as the feedforward compensation for the hysteresis nonlinearity of the SMA.

[0104]

[0105] The second formula, i.e., the inverse model of the phase change hysteresis module, is the inverse of the Duhem differential model, where δ>0 is an arbitrarily small constant to avoid calculation overflow caused by a zero denominator. The meanings of the remaining symbols are consistent with those of the positive model in step S1.

[0106] In step S3, an adaptive sliding mode controller based on proportional-integral-derivative setting is designed, and the control quantity u is composed of the sliding control signal and the sliding mode equivalent control signal: u = u s +u eq .u sis the sliding control signal, which switches the system to the sliding surface; u eq It is a sliding mode equivalent control. A proportional-integral-differential tuning controller is used to approximate the sliding mode equivalent control signal, and Lyapunov's law is used to prove the stability of the closed-loop feedback control system. The specific design process is:

[0107] S3.1: The mathematical description of the shape memory alloy actuator is shown in equation (6):

[0108]

[0109] Where x = [x1(t), x2(t)] T represents the displacement and velocity of the end of the shape memory alloy wire in the shape memory alloy actuator; b(x1,x2) is the input gain, which represents the internal resistance of SMA; f(·) is a nominal parameter of the model, Δf(·) represents the phase change of SMA, describing the hysteresis of SMA; d(t) represents the external disturbance; u(t)=I is the input. Δf(·), d(t) and b(·) are all bounded variables, |Δf(·)|≤g, |d(t)|≤α, b min ≤b(·)≤b max ; g is the maximum change of the hysteresis curve; α is the maximum value of the disturbance.

[0110] S3.2: Design of integral sliding surface σ

[0111] In order to improve the convergence performance of the sliding surface, the reference signal x is defined r , so that:

[0112]

[0113] Where e represents the expected displacement y d The error with the output y, e = y d -y; K1 and K0 make s 2 The root of +K1s+K0=0 is in the left half plane of the s plane;

[0114] Design integral sliding surface σ=x r -x2, substituting into equation (7), σ can be rewritten as When sliding mode occurs, σ tends to zero, and the introduction of integral links and high-order forms in the sliding mode function can effectively improve the control performance of the system;

[0115] S3.3: Stability analysis of closed-loop feedback control systems:

[0116] Define the Lyapunov function as V = 1 / 2σ 2 ,but To achieve system stability Design control u to satisfy K>0; combined with formula (6), the sliding surface σ=x r The derivative of -x2 is expressed as:

[0117]

[0118] Therefore, the control quantity is obtained as:

[0119]

[0120] The control quantity u can be divided into two parts: u = u s +u eq .u s is the sliding control signal, which switches the system to the sliding surface. u eq is the sliding mode equivalent control,

[0121] S3.4: Sliding mode equivalent control signal u eq Design

[0122] Use PID tuning controller to approximate equivalent control:

[0123]

[0124] For the PID parameter setting, the following objective function is selected: And use the gradient descent method to obtain the update rule of PID parameters:

[0125]

[0126] Since when ε u →0, there exists σ→0, so the above PID parameter update rule can be optimized as:

[0127]

[0128] S3.5: Sliding control signal u s Design

[0129] From equations (8), (9) and (10), we can deduce:

[0130]

[0131] Among them, u PID is a positive term, so if K is selected to satisfy the following conditions, then the system must be asymptotically stable If and only if σ=0, the equality holds:

[0132]

[0133] In order to weaken the high-frequency chattering inherent in the control input, the saturation function sat(σ / δ) is used to replace sgn(σ), where δ is the width of the boundary layer. The sliding surface function σ with any initial value will reach and remain within the boundary layer |σ|<δ, and the sliding control signal u s becomes:

[0134]

[0135] In step S4, Figure 5 As shown, the control quantities designed in step S2 and step S3 are introduced into the closed-loop system, and a control method combining feedforward and feedback is adopted to precisely control the displacement of the shape memory alloy actuator. The specific steps are:

[0136] S4.1: The output of the inverse model of the shape memory alloy actuator established in step S2 is: the current required by the shape memory alloy actuator to achieve the desired strain, which is input into the positive model as a feedforward signal in the control system to compensate for the effects of the highly nonlinear, large hysteresis and slow cooling hysteresis of the phase change process inside the shape memory alloy on the shape memory alloy actuator.

[0137] S4.2: In step S3, an adaptive sliding mode controller based on proportional-integral-derivative tuning is designed. The control variable u is composed of the sliding control signal and the sliding mode equivalent control signal: u = u s +u eq This part is input into the mathematical model of the shape memory alloy actuator as a feedback control signal.

[0138] S4.3: Combine the forward model in step S1, the inverse model in step S2 and the adaptive sliding mode controller based on proportional-integral-derivative setting in step S3 to construct a control system for closed-loop displacement control of the shape memory alloy actuator, which is used to accurately control the displacement of the shape memory alloy actuator.

[0139] In step S5, according to steps S1, S2, S3, and S4, the designed control system is modeled and simulated using Matlab / Simulink software. By selecting different tracking signals, the output displacement is tracked and tested, and compared with different controllers to observe the effect of displacement control. The specific steps are as follows:

[0140] S5.1: Build a Simulink simulation platform based on the derivation of steps S1, S2, S3, and S4;

[0141] S5.2: The input is a sinusoidal signal, simulating the sinusoidal tracking condition of the shape memory alloy actuator, and observing the tracking of the output displacement of the shape memory alloy actuator.

[0142] S5.3: The input is a continuously decreasing step signal, which simulates the tracking condition of the shape memory alloy actuator to the step command signal during the heating and contraction process, and observes the tracking condition of the output displacement of the shape memory alloy actuator.

[0143] S5.4: The input is a continuous rising step signal, simulating the tracking condition of the shape memory alloy actuator to the step command signal during the recovery (elongation) process, and observing the tracking condition of the output displacement of the shape memory alloy actuator.

[0144] Example:

[0145] The parameters used in the simulation process are listed in Table 1.

[0146] Table 1 Parameters of mathematical model of shape memory alloy actuator

[0147]

[0148]

[0149] The parameters of the SMC-PID controller designed by the present invention are as follows: the sliding surface parameter is K0 = 0.1s -1 , K1 = 10s; PID gain update rate η1 = 1, η2 = 0.2, η3 = 0.1; sliding mode control signal related parameters are K max =0.84A, δ=0.3. For comparative analysis, when the control effects of the classic PID and the feedforward inverse compensation PID are adjusted to the best, the parameters of the control group controller are: K P =10, K I =K D =0.4.

[0150] When the command signals of the shape memory alloy actuator of the present invention are respectively a sine signal, a continuous descending step signal and a continuous ascending step signal, the displacement tracking of the command signal is as follows: Figure 6 , Figure 7 and Figure 8 For comparative analysis, the control effects of the classic PID and feedforward inverse compensation PID in the comparison group are adjusted to the best. Figure 6 , Figure 7 and Figure 8 It can be seen that the modeling and closed-loop displacement control method of the shape memory alloy actuator described in the present invention has the fastest response speed and the smallest tracking error. Therefore, it can be seen from the operating results of the embodiment that the control method designed by the present invention has the best control effect compared with the classic PID and feedforward inverse compensation PID under various working conditions such as reciprocating, continuous heating deformation, and continuous cooling deformation, and achieves strong robustness against system hysteresis and parameter uncertainty.

[0151] It should be noted that the above-described embodiments are only preferred embodiments of the present invention. For those skilled in the art, without departing from the principles of the present invention, the present invention may be modified, improved and replaced with equivalents, and these modifications, improvements and equivalent replacements are also considered to fall within the scope of protection of the claims of the present invention.

Claims

1. A modeling and closed-loop displacement control method for a shape memory alloy actuator, comprising the following steps: S1: A mechanism model of the shape memory alloy actuator is established based on the working principle of the shape memory alloy actuator, and a mathematical model of the shape memory alloy actuator is established in a modular manner according to the mechanism model, namely, a positive model. The positive model takes current as input and shape memory alloy wire strain as output, and includes a heat transfer and temperature module, a phase change hysteresis module, a strain module, and a resistance module; S2: using the forward model to invert and establish an inverse model of the shape memory alloy actuator, the inverse model is derived in a modular manner, with the desired strain as input and the required current calculated by the inverse model as output. Through the inverse model of the shape memory alloy actuator, the hysteresis effects of the highly nonlinear, large hysteresis and slow cooling caused by the phase change process inside the shape memory alloy will be compensated before the shape memory alloy actuator encounters them, and the inverse model is thus used for inverse hysteresis feedforward compensation control; S3: Design an adaptive sliding mode controller based on proportional-integral-derivative tuning as the feedback control part for achieving precise displacement control of the shape memory alloy actuator, where: The control signal is composed of a sliding control signal and a sliding mode equivalent control signal. A proportional-integral-differential tuning controller is used to approximate the sliding mode equivalent control signal, and the stability of the closed-loop feedback control system is proved by using Lyapunov's law. S4: combining the forward model described in step S1, the inverse model described in step S2, and the adaptive sliding mode controller based on proportional-integral-derivative tuning described in step S3 to construct a control system for closed-loop displacement control of the shape memory alloy actuator; S5: Use Matlab / Simulink software to simulate and test the control system described in step S4, and observe the closed-loop displacement control effect of the shape memory alloy actuator by selecting different tracking signals.

2. The modeling and closed-loop displacement control method of the shape memory alloy actuator according to claim 1, characterized in that: The step S1 comprises: S1.1: The shape memory alloy actuator is driven by a spring bias. The shape memory alloy actuator is composed of a bias spring and a shape memory alloy wire. One end of the spring and the shape memory alloy wire are fixed at the fixed end respectively. When the shape memory alloy wire is powered on and heated, its material contracts due to the shape memory effect and generates tension. When the tension exceeds the elastic force of the spring, the displacement of the end of the shape memory alloy wire will change accordingly. When cooled, the shape memory alloy wire expands with the help of the bias spring. The purpose of accurately controlling the displacement of the shape memory alloy actuator is achieved by accurately controlling the current input to the shape memory alloy wire. S1.2: The heat transfer and temperature module is used to describe the heat transfer problem of the shape memory alloy wire in the shape memory alloy actuator. The heat transfer and temperature module model is given by the following expression (1), with current as input and the temperature of the shape memory alloy wire as output: Where R is the internal resistance of the alloy wire, d0 and L0 are the cross-sectional diameter and length of the undeformed alloy wire, respectively, and T amb is the ambient temperature, ρ is the mass density of the alloy wire, c is the specific heat of the alloy wire, h is the convection heat transfer coefficient of the alloy wire, c and h are functions of the temperature T; S1.3: The phase change hysteresis module is used to describe the martensite volume fraction R of the shape memory alloy wire in the shape memory alloy actuator. m The relationship with temperature T is given by the Duhem differential hysteresis model. The phase change hysteresis model is given by the following expression (2): Among them, g + / - is the Gaussian probability density function, describing the slope of the hysteresis loop, h + / - It is for g + / - The integral of describes the shape of the hysteresis loop. The subscripts + and - represent increasing and decreasing curves, respectively. μ is the mean value, and σ 2 is the variance, which is used to control the shape of the hysteresis loop. is the error function when integrating the normal distribution; S1.4: The strain module is used to describe the strain ε and martensite volume fraction R of the shape memory alloy wire in the shape memory alloy actuator. m The relationship between R and ε is given by the following formula (3). The total strain ε can be approximated as m Polynomial of: Among them, ε0 is the strain caused by the preload of the alloy wire in the fully austenitic state, and the values ​​of k1, k2, and k3 are related to the specific shape memory alloy wire; S1.5: The resistance module is used to describe the resistance of the shape memory alloy wire in the shape memory alloy actuator, and is provided to step S1.2 for calculation, which is given by the following formula (4): the temperature T and the martensite volume fraction R calculated by steps S1.2, S1.3 and S1.4 are: m The resistance of the shape memory alloy wire can be calculated by the strain ε: Among them, ρ a (T) is the resistivity of austenite, ρ m (T) is the resistivity of martensite, p1, p2, p3, q1, q2 and α i (i=1-9) is the correlation constant.

3. The modeling and closed-loop displacement control method of the shape memory alloy actuator according to claim 1, characterized in that: The step S2 comprises: Assume the desired strain trajectory is ε rd (t), according to the forward model of the shape memory alloy driver, the inverse models of the strain module, the phase change hysteresis module, the heat transfer and temperature module and the resistance module are obtained by sequentially inverting, as shown in the following formula (5), thereby obtaining the inverse model of the shape memory alloy driver whose input is the desired strain and whose output is the required current calculated by the inverse model. Through the inverse model of the shape memory alloy driver, the hysteresis effects of the highly nonlinear, large hysteresis and slow cooling caused by the phase change process inside the shape memory alloy will be compensated before the shape memory alloy driver encounters them, and the inverse model is thus used for inverse hysteresis feedforward compensation control; Among them, δ>0 is an arbitrarily small constant to avoid calculation overflow caused by the denominator being zero.

4. The modeling and closed-loop displacement control method of the shape memory alloy actuator according to claim 1, characterized in that: The step S3 comprises: Design an adaptive sliding mode controller based on proportional-integral-derivative tuning. The control variable u is composed of the sliding control signal and the sliding mode equivalent control signal: u = u s +u eq , where u s is the sliding control signal, which switches the system to the sliding surface; u eq It is a sliding mode equivalent control; a proportional-integral-differential tuning controller is used to approximate the sliding mode equivalent control signal, and the Lyapunov law is used to prove the stability of the closed-loop feedback control system. The specific design process is: S3.1: Mathematical description of shape memory alloy actuators: Where x = [x1(t), x2(t)] T represents the displacement and velocity of the end of the shape memory alloy wire in the shape memory alloy actuator; b(x1, x2) is the input gain, which represents the internal resistance of the shape memory alloy; f(·) is a nominal parameter of the model, Δf(·) represents the phase change of the shape memory alloy, describing the hysteresis phenomenon of the shape memory alloy; d(t) represents the external disturbance; u(t) = I is the input; Δf(·), d(t) and b(·) are all bounded variables, |Δf(·)|≤g, |d(t)|≤α, b min ≤b(·)≤b max ; g is the maximum change of the hysteresis curve; α is the maximum value of the disturbance; S3.2: Design of integral sliding surface σ In order to improve the convergence performance of the sliding surface, the reference signal x is defined r , so that: Where e represents the expected displacement y d The error with the output y, e = y d -y; K1 and K0 make s 2 The root of +K1s+K0=0 is in the left half plane of the s plane; Design integral sliding surface σ=x r -x2, substituting into equation (7), σ can be rewritten as When sliding mode occurs, σ tends to zero, and the introduction of integral links and high-order forms in the sliding mode function can effectively improve the control performance of the system; S3.3: Stability analysis of closed-loop feedback control systems: Define the Lyapunov function as V = 1 / 2σ 2 ,but To achieve system stability Design control u to satisfy K>0; combined with formula (6), the sliding surface σ=x r The derivative of -x2 can be expressed as: Therefore, the control quantity is obtained as: The control quantity u can be divided into two parts: u = u s +u eq ;u s is the sliding control signal, which switches the system to the sliding surface. u eq is the sliding mode equivalent control, S3.4: Sliding mode equivalent control signal u eq Design: Use PID tuning controller to approximate equivalent control: For the PID parameter setting, the following objective function is selected: And use the gradient descent method to obtain the update rule of PID parameters: Since when ε u →0, there exists σ→0, so the above PID parameter update rule can be optimized as: S3.5: Sliding control signal u s Design: From equations (8), (9) and (10), we can deduce: Among them, u PID is a positive term, so if K is selected to satisfy the following conditions, then the system must be asymptotically stable If and only if σ=0, the equality holds: In order to weaken the high-frequency chattering inherent in the control input, the saturation function sat(σ / δ) is used to replace sgn(σ), where δ is the width of the boundary layer. The sliding surface function σ with any initial value will reach and remain within the boundary layer |σ|<δ, and the sliding control signal u s becomes:

5. The modeling and closed-loop displacement control method of the shape memory alloy actuator according to claim 1, characterized in that: The step S4 comprises: S4.1: The output of the inverse model of the shape memory alloy actuator established in step S2 is: the current required by the shape memory alloy actuator to achieve the desired strain, which is input into the forward model as a feedforward signal in the control system to compensate for the effects of the highly nonlinear, large hysteresis and slow cooling hysteresis of the phase change process inside the shape memory alloy on the shape memory alloy actuator; S4.2: In step S3, an adaptive sliding mode controller based on proportional-integral-derivative tuning is designed. The control variable u is composed of the sliding control signal and the sliding mode equivalent control signal: u = u s +u eq ; This part is input into the mathematical model of the shape memory alloy actuator as a feedback control signal; S4.3: Combine the forward model described in step S1, the inverse model described in step S2, and the adaptive sliding mode controller based on proportional-integral-derivative setting described in step S3 to construct a control system for closed-loop displacement control of the shape memory alloy actuator, which is used to accurately control the displacement of the shape memory alloy actuator.

6. The modeling and closed-loop displacement control method of the shape memory alloy actuator according to claim 1, characterized in that: The step S5 comprises: S5.1: According to the derivation of steps S1, S2, S3, and S4, a Simulink simulation platform is built; S5.2: The input is a sinusoidal signal, simulating the sinusoidal tracking condition of the shape memory alloy actuator, and observing the tracking of the output displacement of the shape memory alloy actuator; S5.3: The input is a continuous descending step signal, simulating the tracking condition of the shape memory alloy to the step command signal during the heating and contraction process, and observing the tracking condition of the output displacement of the shape memory alloy driver; S5.4: The input is a continuous rising step signal, which simulates the tracking condition of the shape memory alloy to the step command signal during the recovery process, and observes the tracking condition of the output displacement of the shape memory alloy driver.

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